# Can we expand matrices in a Taylor series?

Consider a matrix $n\times n$ $A$ which depends on a $n\times n$ matrix $B$ i.e., $A=A(B)$. The matrix $B$ in turn depends upon a continuous real parameter $\phi$ such that for small $\phi$, $B$ can be written as $$B(\delta\phi)=\mathbb{1}-C\delta\phi$$ where $C$ is also a $n\times n$ matrix.

Can we expand $A(B(\delta\phi))=A(\mathbb{1}-C\delta\phi)$ in a Taylor series?

I would simplify your question in this way: how a matrix with entries $a_{ij}=a_{ij}(t)$ depending on a real parameter $t$ can be expanded into a Taylor series ?

There is a simple answer : expand separately all the $a_{ij}(t)$s into Taylor series (around the origin or around another point, if they can be expanded...), then factorize $1,t,t^2...$.

The best is to see it on an example :

$$\begin{pmatrix}\cos(t)&\sin(t)\\0&1\end{pmatrix}=I_2+t\begin{pmatrix}0&1\\0&0\end{pmatrix}+t^2/2\begin{pmatrix}-1&0\\ \ \ 0&0\end{pmatrix}+\cdots$$

• Actually, I need to know the expansion of something like $A(I+C\epsilon)$ where A, C are square matrices, and I is the identity matrix, and $\epsilon$ is small. It's actually related to a physics question I asked here. physics.stackexchange.com/questions/398463/… @JeanMarie
– SRS
Apr 8, 2018 at 8:41
• I have had a look at your question on physics SE. I think that finding higher order terms deals with Baker-Campbell-Hausdorff formula involving brackets, brackets of brackets, etc.(en.wikipedia.org/wiki/…). Besides : what you call "Wigner's theorem" is not known in mathematics under this terminology ; it is a fact that any rotation matrix can be expressed as the exponential of a skew symmetric matrix of the same size with coefficients that are linked to a normalized axis and the rotation angle. Apr 8, 2018 at 8:59
• @JeanMarie how would one generalise this expansion to for example two variables ? Say $a_{ij} = a_{ij}(t, u)$ Jan 22, 2019 at 13:42